Isogeometric Analysis on Implicit Domains: Approximation, Stability and Error Estimates

Abstract

The gap between finite element analysis and computer-aided design derives the development of isogeometric analysis (IGA), which uses the same representation in the geometry and the analysis. However, the parameterization in IGA is non-trivial. Weighted extended B-splines (WEB) method replaces grid generation and parameterization with weight function construction (R-function or distance function). By using implicit spline representation, isogeometric analysis on implicit domains (IGAID) adopts the merits of the “isoparametric” in IGA and “weight function generation” in WEB. But the theoretical properties have not been fully studied yet. In this paper, we study the theoretical aspects of IGAID using tensor-product B-splines. Both the approximation and stability properties of IGAID are considered. By setting appropriate constraints on the weight function, we can derive the optimal approximation order and stability. Numerical examples show the effectiveness of the approach and validate the theoretical results.

8 Appendix

1.1 A. The Proof of Theorem 4.1

Proof

We prove Eq. 4.2 first.

By Eq. 3.2, $B_i=\frac{w(x)}{w(x_i)}{\tilde{b}}_i=\frac{w(x)}{w(x_i)}(b_i+\sum \limits _{j\in J(i)}e_{ij}b_j),$ we have

$$\begin{aligned}\begin{aligned} ||B_i||_{0,\tilde{Q_i}}&\le \left( \max \limits _{\tilde{Q_i}}\frac{w(x)}{w(x_i)}\right) \left\| b_i+\sum \limits _{j\in J}e_{ij}b_j\right\| _{0,\tilde{Q_i}}\\&\preceq \max \limits _{\tilde{Q_i}}\left| \frac{w(x)}{w(x_i)}\right| . \end{aligned}\end{aligned}$$

Then, we prove Eq. 4.3,

$$\begin{aligned} DB_i=\frac{Dw}{w(x_i)}\left( b_i+\sum \limits _{j\in J}e_{ij}b_j\right) +\frac{w}{w(x_i)}\left( Db_i+\sum \limits _{j\in J}e_{ij}Db_j\right) , \\ |B_i|_{1,\tilde{Q_i}}=||DB_i||_{0,\tilde{Q_i}}\preceq \max \limits _{\tilde{Q_i}}\frac{|Dw|}{w(x_i)}+h^{-1}\max \limits _{\tilde{Q_i}}\frac{w(x)}{w(x_i)}. \end{aligned}$$

We have

$$\begin{aligned} \begin{aligned} \left\| DB_i\right\| _{1,\tilde{Q_i}}&=\left( \left\| B_i\right\| ^2_{0,\tilde{Q_i}}+|DB_i|^2_{1,\tilde{Q_i}}\right) ^{1/2}\\&\le \left\| B_i\right\| _{0,\tilde{Q_i}}+|DB_i|_{1,\tilde{Q_i}}\\&\preceq \frac{1}{w(x_i)}\max \limits _{\tilde{Q_i}}\left[ (1+h^{-1})w(x)+|Dw|\right] . \end{aligned} \end{aligned}$$

Last, we prove Eq. 4.4

$$\begin{aligned}\begin{aligned} \left\| \wedge _i\right| _0&=\left\| \frac{w(x_i)}{w(x)}\lambda _i\right\| _0\le \max \limits _{x\in Q^{\prime }_i}\frac{w(x_i)}{w(x)}||\lambda _i||_0\\&\preceq \max \limits _{x\in Q^{\prime }_i}\frac{w(x_i)}{w(x)}. \end{aligned}\end{aligned}$$

$\square$

1.2 B. The Proof of Theorem 4.2

Proof

We only give the proof of Eq. 4.5, Eq. 4.6 can be proved similarly.

$$\begin{aligned} \left\| \sum \limits _{i \in I}\alpha _iB_i\right\| ^2_0\le \sum _{i\in I}\left( |a_i|^2||B_i||_0^2\right) \le \left( \max _{i\in I}||B_i||^2_0\right) \sum _{i\in I}|a_i|^2\le \left( \max _{i\in I}||B_i||^2_0\right) ||A||^2. \end{aligned}$$

Therefore, $\left\| \sum \limits _{i \in I}\alpha _iB_i\right\| _0\le \left( \max \limits _{i\in I}||B_i||_0\right) ||A||.$ $\square$

1.3 C. The Proof of Theorem 4.3

Proof

Let $wp = \sum \limits _{i\in I} \alpha _i B_i$, times $\wedge _j$ and integral lead to

$$\begin{aligned} \int _{\Omega } wp\wedge _j = \sum \limits _{i\in I} \alpha _i \int _{\Omega } \wedge _j B_i, \end{aligned}$$

then

$$\begin{aligned} \alpha _j=\int _{\Omega } wp\wedge _j. \end{aligned}$$

$\square$

1.4 D. The Proof of Lemma 5.1

Proof

We prove the first inequation.

$\frac{w(x)}{w(x_i)}=1+\frac{w(x)-w(x_i)}{w(x_i)}$, by the smoothness of w and the boundness of $|\nabla w|$, we have

$$\begin{aligned} |w(x)-w(x_i)|\le c |x-x_i|\preceq h,\qquad by \ |x-x_i|\preceq h. \end{aligned}$$

If $\textrm{dist}(x_i,\partial \Omega )>\delta ,$ using Fact 2, we have $w(x_i)\ge c_3,$ therefore,

$$\begin{aligned} \left| \frac{w(x)}{w(x_i)}\right| \le 1+\left| \frac{w(x)-w(x_i)}{w(x_i)}\right| \preceq 1+\frac{h}{c_3}\preceq 1. \end{aligned}$$

If $\textrm{dist}(x_i,\partial \Omega )\le \delta ,$ using Fact 1, we have

$$\begin{aligned} w(x_i)\asymp \textrm{dist}(x_i,\partial \Omega )\succeq h, \qquad \mathrm {by\ dist}(x_i,\partial \Omega )\succeq h. \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \frac{w(x)}{w(x_i)}\right| \le 1+\left| \frac{w(x)-w(x_i)}{w(x_i)}\right| \preceq 1+\frac{h}{ h}\preceq 1. \end{aligned}$$

Then, we prove the second inequation.

As $x\in \sup (\wedge _i),$ we have $\hbox {dist}(x,\partial \Omega )\succeq h$. Similar discussion as Lemma 5.1, we have

$\frac{w(x_i)}{w(x)}=1+\frac{w(x_i)-w(x)}{w(x)}$, by the smoothness of w and the boundedness of $|\nabla w|$, we have

$$\begin{aligned} |w(x)-w(x_i)|\le c |x-x_i|\preceq h,\qquad by \ |x-x_i|\preceq h. \end{aligned}$$

If $\hbox {dist}(x,\partial \Omega )>\delta ,$ using Fact 2, we have $w(x)\ge c_3.$ Therefore,

$$\begin{aligned} \left| \frac{w(x_i)}{w(x)}\right| \le 1+\left| \frac{w(x)-w(x_i)}{w(x)}\right| \preceq 1+\frac{h}{c_3}\preceq 1. \end{aligned}$$

If $\hbox {dist}(x,\partial \Omega )\le \delta ,$ using Fact 1, we have

$$\begin{aligned} w(x)\asymp \hbox {dist}(x,\partial \Omega )\succeq h. \qquad \hbox {by} \ \hbox {dist}(x,\partial \Omega )\succeq h \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \frac{w(x_i)}{w(x)}\right| \le 1+\left| \frac{w(x)-w(x_i)}{w(x)}\right| \preceq 1+\frac{h}{ h}\preceq 1. \end{aligned}$$

$\square$